Cabriolet for RISC OS

Transcription

1 abriolet for RIS OS a dynamic geometry program Version 2.5 i

2 opyright opyright Murklesoft Murklesoft can be contacted care of: Icon Technology hurch House hurch Street arlby Stamford Lincs PE9 4N +44 (0) Upgrades to abriolet (as a zip d archive) and the latest version of this guide (in TechWriter and PF format) are available from: redits This guide was written and typeset on RIS OS using TechWriter Professional. ll the diagrams were drawn in abriolet and exported to TechWriter in raw format. abriolet was written by Michael orcherds. First Published October 1997 by Murklesoft Manual Issue October 2002 ii abriolet for RIS OS:

3 ontents opyright ii redits ii bout this Guide v onventions used v hapter one Using abriolet The Toolbar The uttons Tool Menus The Panes Menus The Mouse uttons The Mouse Pointer Reconfiguring the Menus hapter two Tutorial Introduction ircumcircle of a Triangle Incircle of a Triangle Tracing a Point hapter three Reference Point and Label Tool Intersection Tool Midpoint Tool Perpendicular Point Tool Mirror Point Tool Rotate Point Tool Vector Point Tool Line Segment Tool Line Tool ircle Tool ircle Given Radius Tool Ellipse Tool Perpendicular Line Tool Parallel Line Tool Perpendicular isector Tool ngle isector Tool ngle Trisector Tool Measurement Tool Measure rea Tool Mark ngle Tool Hide Object Tool Thicken Line Tool olour Red Tool olour Grey Tool olour lack Tool olour Purple Tool olour lue Tool iii

4 elete Object Tool Set Ratio Tool Link Objects Tool nimate Tool Static Locus Tool Vector Tool Vector Sum Tool opy Vector Tool ddition Tool Subtraction Tool Multiplication Tool Square Tool Square Root Tool X oordinate Tool Y oordinate Tool iv abriolet for RIS OS:

5 bout this Guide This guide is about abriolet a dynamic geometry package for that has been designed both to be easy to use and to run at a good speed even on low specification machines. The manual is divided into three chapters: hapter one describes how abriolet works. hapter two is a tutorial. hapter three is a reference section detailing all the Tools that abriolet has. onventions used ragging the mouse between two points means to press the appropriate mouse button, moving the mouse and then finally releasing the button again.where click is used on its own it refers to the left hand mouse button. Geometrical objects that abriolet uses are referred to in italic, for example Perpendicular isector. Tools are referred to in bold, for example, Perpendicular isector Tool. v

6 vi abriolet for RIS OS:

7 hapter one Using abriolet hapter one Using abriolet 1

8 abriolet The Toolbar To load abriolet, double-click on it from a filer window. abriolet s icon will then appear on the icon bar. lick on this icon each time you wish to open a new abriolet document. The toolbar is at the top of every abriolet window. It consists of two parts: the buttons on the bottom left are for selecting different tools and on the top and right are two panes which display helpful information. The uttons Tool Menus s you move the mouse over the buttons, the top pane tells you what each Tool is. licking with Select (the left mouse button) will then activate that tool (see hapter three for a thorough description of each Tool). Each button can be used to select more than one Tool. To select a different Tool to the one displayed in the pane, press the middle mouse button Menu as instructed in the lower pane. This brings up a menu of other available Tools which can be chosen. This will then be the default Tool for that button, until another Tool is chosen from its menu. The Panes The two panes display different information depending upon where the mouse pointer is. When the mouse pointer is over the buttons on the Tool ar, the top pane gives you the name of each Tool. When the mouse pointer is over the main window, the top pane tells you how to use the current Tool. The bottom pane will usually give some information about the object that the mouse pointer is over. Menus part from the menus used to select different Tools, there are two other menus available. The first is obtained by pressing Menu (the middle mouse button) over abriolet s icon on the icon bar. This allows you to either quit abriolet or view some information about abriolet, including the version number and who is licensed to use it. The second is obtained by pressing Menu over abriolet s main window. This allows you to: Save a file, so that it can be reloaded on a future occasion (also F3). Export a file in raw format so that it can be used in other programs. Save a locus out as a SV file. Show any objects that may have been hidden. Undelete any objects that may have been deleted. Turn the grid on or off, and set its size. Undo or Redo the last action (also F8). 2 abriolet for RIS OS: hapter one Using abriolet

9 The Mouse uttons Select, the left button, is generally used for selecting an object, or the position for an object (depending on the Tool selected). For example when using the Line Segment Tool, you click and drag from where you want the line to start. If you click near a point, the Line Segment will start at this point. However if you click nowhere near a point then a starting point will be automatically created. Similarly if you place the mouse near a Point when the left hand mouse button is pressed, the Line Segment will snap onto that Point. If you release the mouse button here, the Line Segment will have this Point as its endpoint. The right button can always be used for dragging points and labels around (although not all points and labels can be moved). The Mouse Pointer Reconfiguring the Menus The mouse pointer will change depending on what object it is near. This is useful when two objects are close together and you need to know which one you are clicking on. If you press and hold lt when dragging the mouse, it will confine the point being dragged to a grid. You can also turn the grid on permanently (see Menu section, above). ll the the Tool menus obtained by pressing Menu over the Tool buttons can be reconfigured, as can the buttons themselves. To change them, load the file!abriolet.efault.menus into a text editor such as Edit or Zap. Each line of the file defines the menu for a button. It consists of names of Tools as they appear in the menus, separated by commas. Up to fifteen buttons can be defined. When the file has been altered it should be saved in the appropriate directory for your configured country (e.g.!abriolet.efault.uk) which you can find out by pressing F12 and typing: Show abriolet$ountry If the menus are reconfigured then they will obviously not be as described in the manual! hapter one Using abriolet 3

10 4 abriolet for RIS OS: hapter one Using abriolet

11 hapter two Tutorial hapter two Tutorial 5

12 Introduction This chapter is a short tutorial that shows you, step by step, how to create the following examples: ircumcircle of a Triangle Incircle of a Triangle Tracing a Point 6 abriolet for RIS OS: hapter two Tutorial

13 ircumcircle of a Triangle The circumcircle of a triangle always passes through all its corners. 1 Open a new document (after loading abriolet) by clicking on its icon on the icon bar. 2 Now construct Line Segment. Press the right hand mouse button over the second button on the Toolbar, which will bring up a menu. hoose Line Segment from this. Then, over the main window, hold down the left mouse button and move the mouse before releasing it. 3 Then construct a triangle. Move the mouse to the Point (the mouse pointer changes to say This Point ). Then hold down the left hand mouse button and move the mouse to where you wish the Point to be, before releasing the mouse button. 4 Now complete the triangle. Move the mouse to Point, press the left hand mouse button, move the mouse to Point and release the mouse button. efore proceeding, check that the Line Segments join up. Hold down the right hand mouse button over any of, or and then move the mouse around: the Point that you have clicked on should follow the mouse around, as should the Line Segments joined to that Point. 1 Now construct two Perpendicular isectors. lick the right hand mouse button over the fourth button from the left on the Toolbar. hoose Perpendicular isector from the menu. Then simply click on Line Segment and then on Line Segment. 2 Now construct the Intersection of the Perpendicular isectors. lick the right hand mouse button over the leftmost button and choose Intersection. Then click on both the Perpendicular isectors in turn. 3 Now construct the circumcircle itself. hoose ircle from the Toolbar (it is the default tool for the third button, so simply click on it, or you can choose it from the menu obtained by clicking the right hand mouse button). Then move the mouse pointer to Point, hold down the left hand mouse button, move the mouse to Point and then release the mouse button. hapter two Tutorial 7

14 4 If you wish you can now hide the construction lines. hoose Hide Object (seventh button) and then click on any lines, points or labels that you don t want to be shown. 5 Now check that the circumcircle always goes through Points, and by dragging these three points round with the right hand mouse button. 8 abriolet for RIS OS: hapter two Tutorial

15 Incircle of a Triangle The incircle of a triangle always touches the edges of the triangle. 1 Start off by opening up a new document. 2 s before, start off by constructing a triangle from three Line Segments. 3 Next construct the ngle isector of angle. hoose ngle isector (from the menu on the fourth button) then click on the three Points,, in that order. 4 Then construct the ngle isector of angle. Still using the ngle isector Tool click on the three Points,, in that order. 5 Then choose Intersection (from the first button) to construct the Intersection of the ngle isectors by clicking on each one in turn. 6 Now to construct a Point on the Line Segment that is on the radius of the incircle, we use the Perpendicular Point Tool (from the menu of the first button). lick on Point and then on Line Segment. E 7 Finally we can construct the incircle using the ircle Tool (third button). lick over Point (the centre of the ircle), move the mouse pointer to Point E (the radius point) and then release the mouse button. E hapter two Tutorial 9

16 8 heck that the incircle always touches all three sides of the triangle by dragging the corners around with the right hand mouse button. E 10 abriolet for RIS OS: hapter two Tutorial

17 Tracing a Point Here we will construct a Point whose locus is an ellipse (i.e. it will trace out an ellipse when we move another point). 1 First use the Line Segment Tool (second button s menu) to make Line Segment. 2 Next use the Point and Label Tool (first button) to construct Point on Line. heck by dragging it with the right hand mouse button that it cannot move off Line Segment. 3 To construct the ircle centre we use the ircle Given Radius Tool (third button s menu). To use it, first click where you wish Point to be, then click on Point, then on Point. Now the ircle s radius will always be the same length as the line joining and. heck this by dragging Point with the right hand mouse button: the ircle should change size as Point moves. 4 Similarly, still using the ircle Given Radius Tool, construct the second ircle by clicking where you wish Point E to be, then on Point and then on Point. E F G 5 Now use the Intersection Tool (first button s menu) to construct Points F and G. Simply click on the first ircle, and then the second. E hapter two Tutorial 11

18 6 Now as Point is moved along Line Segment (with the right hand mouse button) Points F and G will both trace out half an ellipse. F E G To see this, choose Static Locus (seventh button s menu). Then click on, say, Point F. Then drag Point with the right hand mouse button. The trail left will be traced out in Green. To see the path traced out by Point G, click on Point G with the left hand mouse button and drag Point with the right hand mouse button. nother way to check this (as shown) is to actually construct the Ellipse. To do this, choose Ellipse (third button s menu) and then click on Points, E and F in that order (so that and E are the foci of the Ellipse and F is a Point on its edge). Then Points F and G will stay on the Ellipse as Point is moved. 12 abriolet for RIS OS: hapter two Tutorial

19 hapter three Reference hapter three Reference 13

20 Tools The tools can be choosen from the utton bar. Pressing the right hand mouse button over a button will reveal additional tools. This brings up a menu of other available Tools which can be chosen. This will then be the default Tool for that button, until another Tool is chosen from its menu. Point and Label Tool To create a asic Point, click anywhere away from other objects. The Point can then be dragged to where it is required with the right hand mouse button. Label is automatically created next to the Point. It can also be moved with the right hand mouse button. To create a Point on ircle or a Point on Line, move the mouse until the pointer changes to indicate that you are near enough to a ircle or any sort of line (Line, Line Segment, ngle isector, Parallel Line, Perpendicular Line or Perpendicular isector) and click with the left hand mouse button. When you then try to move one of these with the right hand mouse button, it will be constrained to move only along the circle or line. is a Point on Line on Line Segment Mathematical efinition point is a geometrical object with zero dimension. Intersection Tool To create the Point(s) at the intersection of two objects, simply click on them. Suitable such objects are ircles and any sort of Line, for example Line Segments, ngle isectors, Perpendicular isectors. E and F are the intersections of Line Segment and the ircle centre, E F Mathematical efinition Midpoint Tool The intersections of two objects are the points which satisfy the conditions for both objects simultaneously. To create the Midpoint of two Points either click on both Points or, if it exists, click on a Line Segment joining them. is the Midpoint of Line Segment Mathematical efinition The midpoint of two points is exactly half way in between them. 14 abriolet for RIS OS: hapter three Reference

21 Perpendicular Point Tool To create the nearest Point on a line to a given Point, click on both the Point and the line. E is the Perpendicular Point obtained by clicking on Point and Line Segment. It has been used to construct the incircle of triangle. ( is the Intersection of the ngle isectors of the triangle) E Mathematical efinition Mirror Point Tool The perpendicular point is such that the line joining the two points is at right angles to the original line. To create a Mirror Point that is the reflection of another point in a line click on both the point and the line. is the Mirror Point of in the line. Hence will always be a kite. Rotate Point Tool The Rotate Point Tool can be used to rotate a Point through a fixed angle. To rotate a whole shape each Point that determines the shape can be rotated and then the shape redrawn. It can also be used to create an angle of a fixed size. To rotate a Point by a fixed angle, first click on the Point to be rotated then click on the Point that is to be the centre of the rotation. window will then appear, into which you should type an angle in degrees. This will be the size of the rotation (anti-clockwise). negative angle can be entered for a clockwise rotation. Point has been rotated through 72 about Point to make Point. Point has been rotated through 72 about Point to make Point. Point has been rotated through 72 about Point to make Point E. Point E has been rotated through 72 about Point to make Point F E F Hence shape EF is a regular pentagon. hapter three Reference 15

22 Vector Point Tool To create a Vector Point, click on a Point and, while still holding the mouse button, move the mouse to where you want the Vector Point and then release the mouse button. The Vector Point will then always remain the same distance and direction from the original Point. is a Vector Point depending upon Point. The ircle has centre and radius point. When Point is moved, Vector Point will also move and so the ircle will stay the same size, rather than growing or shrinking. Note: This tool was called the Vector Tool with a different icon in some previous releases of abriolet. The Vector Tool is similar but can automatically create the initial point and draw a Line Segment. Line Segment Tool To create a Line Segment, drag the mouse with the left hand mouse button pressed from the start point to the end point. If you wish it to start and/or end on a Point, simply start and/or end your drag on the appropriate Point(s). Line Segment Mathematical efinition Related Tools line segment is a finite section of a line, as opposed to an infinite (unending) line which is normally just referred to as a line. Line Tool, Point and Label Tool Line Tool To create a Line, drag the mouse, with the left hand button pressed, from one point you wish to be on the line to another point on the line. If you wish it to go through one or two existing Points, simply start and/or end your drag on the appropriate Point(s). Line produced. 16 abriolet for RIS OS: hapter three Reference

23 ircle Tool To draw a ircle, drag the mouse from where you want the centre to be to where you want the edge of the ircle to be. While you drag the mouse, the ircle will be shown in the position according to the current mouse position. If you want the ircle either to have a particular Point at its centre or to have a particular Point on its radius, simply start or finish the mouse drag on the appropriate Point(s). ircle with radius and radius point. Mathematical efinition ircle Given Radius Tool circle is the set of points equidistant from the centre point. To create a ircle whose radius is the same length as a Line Segment, firstly click on the Point you wish to be the centre then click on the two endpoints of the Line Segment. s with the ircle Tool, if you click where there is no point, one will be automatically created. is a Point on Line Segment. ircle centre has its radius set to Line Segment. ircle centre E has its radius set to Line Segment. F and G are the Intersections of the two ircles. s Point is moved along Line Segment, each of F and G trace out half an ellipse (whose foci are and E). G E F This is because the length G + GE is always the same (and is equal to length + = ), and it is equivalent to the mechanical construction of an ellipse by putting two pins in a piece of paper (at and E), tying a piece of string to them (length ) and moving a pencil round (at Point F or G) keeping the string taught. hapter three Reference 17

24 Ellipse Tool To create an Ellipse, click on three points: two foci and a point that is to be on the circumference. s with the ircle Tool, if you click where there is no point, one will be automatically created. and are the foci of the ellipse, defines where the circumference of the ellipse is. Mathematical efinition n ellipse is a stretched circle. Ellipses have the property that if and are the foci and is any point on the edge then the angle bisector of is perpendicular to the tangent to the ellipse at. Hence if a snooker ball is hit in any direction from one focus, it will always go through the other focus, where a pocket can be cunningly placed! Perpendicular Line Tool To create a line that is perpendicular (at right angles to) another line, click (in either order) on the line you wish to be at right angles to and a Point which you want your new Perpendicular Line to go through. The Perpendicular Line through will always go through and will always remain at right angles to Line. Parallel Line Tool To create a line that is parallel to another line, click (in either order) on the line you wish to be a parallel to and a Point which you want your new Parallel Line to go through. The Parallel Line through will always go through and will always remain parallel to Line. 18 abriolet for RIS OS: hapter three Reference

25 Perpendicular isector Tool To create the perpendicular bisector of two Points, click on the first Point (which will turn red) then click on the second Point. To create the Perpendicular isector of a Line Segment, just click on the Line Segment. The Perpendicular isector of. Mathematical efinition perpendicular bisector of two points is a line, each point of which is the same distance (equidistant) from the two original points. In practice this is a line that cuts the Line Segment at right angles exactly half way in between the two points. ngle isector Tool To create an ngle isector, either click on an ngle that has already been marked with the Mark ngle Tool, or click on three Points that define the angle to be bisected. The ngle isector of ngle. ngle Trisector Tool To create an ngle Trisector, either click on an ngle that has already been marked with the Mark ngle Tool, or click on three Points that define the angle to be trisected. The ngle Trisector of angle. hapter three Reference 19

26 Measurement Tool To measure the distance between two Points, click on them or, if it exists, click on a Line Segment joining them. To Measure an angle, it must first have been marked with the Mark ngle tool. Then simply click on the angle. To Measure an area, use the Measure rea tool. alculations can be performed on Measurements using various other Tools (see ddition Tool). Length = 3.79cm, angle = Measure rea Tool To measure the area of a triangle, click on all three corners. The area of triangle is cm Mark ngle Tool To mark an acute angle click on three Points that specify the angle with the second Point being the Point at the angle. ngle has been marked. Uses Hide Object Tool This is necessary for specifying an angle to measure and also allows you to specify an angle more quickly for the ngle isector Tool. It also makes clear which angle is being bisected or measured. To hide an object, simply click on it. Unlike eleting an object, hiding it does not affect any objects that it depends upon (for example if you hide a Line, any Points on that Line will not disappear). To unhide objects, choose Show Hidden from the main menu. Point is a Point on Line on Line Segment, but Line Segment has been hidden. 20 abriolet for RIS OS: hapter three Reference

27 Thicken Line Tool The Thicken Line Tool can be used to draw Lines and ircles with a thicker (bolder) line. To thicken a Line or a ircle, simply click on it. To revert to normal, simply click again. olour Red Tool To colour an object red, simply click on it. olour Grey Tool To colour an object grey, simply click on it. olour lack Tool To colour an object black, simply click on it. olour Purple Tool To colour an object purple, simply click on it. olour lue Tool To colour an object blue, simply click on it. elete Object Tool Set Ratio Tool To delete an object, simply click on it. This will also delete all the objects that depend upon the deleted object. The Hide Object Tool can be used if you simply want an object not to be displayed. ll the objects that have been deleted previously can be recreated by choosing Undelete from the main menu. This works even after a file has been saved and reloaded. lso note that if an object is inadvertently deleted, the Undo/Redo menu option (or pressing F8) can be used to bring back only the just-deleted objects. The Set Ratio Tool can be used to set precisely how far along a Line Segment a Point on Line is. To set this, click on a Point on Line. window will appear into which you should type a decimal value between 0 and 1. This ratio is always preserved, unless the Point on Line is dragged by the mouse. Point on Line has been set to be 0.33 along Line Segment. hapter three Reference 21

28 Link Objects Tool This Tool can be used to change any point into a Point on Line or a Point on ircle. It can even be used to move a Point On ircle/line to a different circle/line. To achieve this, simply click on the desired point to change, and then click on the line or circle that you wish it to lie on. Note that if the Point originally depended upon some other objects, these dependencies will be lost when the Point is redefined. nimate Tool The nimate Tool can be used to create simple animations. To use it, either click on a Point on Line or a Point on ircle. The Point clicked upon will then move smoothly around the ircle or back and forth along the Line. Simpler but more general animations can be created by pressing or holding down F8 to repeatedly undo and then redo the last action taken. Static Locus Tool To create the Locus of a Point (i.e. to trace its path), click on a Point (which will turn red). Then simply drag any other Points about with the right hand mouse button. If the original Point moves, its path will be traced out by green dots. If the mouse is moved slowly and carefully (press Shift before dragging to facilitate this) then the green dots will be closer together so that the Locus will look like a curve. Only one Locus is allowed (including in different windows) at a time. To clear a Locus, select a different Tool and then reselect the Locus Tool. It is possible to save out the coordinates of all the points in the Locus as a omma Separated Variable (SV) file from the menu which is obtained by pressing the middle mouse button over the main window. Peaucellier s ell is a mechanical construction that will draw a line without a straight edge. ll the Line Segments in the diagram have been arranged to have a fixed length (representing rods). s Point moves around the ircle, Point K traces out a line (i.e. the locus of K is a line). K Mathematical efinition The locus of a point is the path traced out by it as it moves under certain constraints. 22 abriolet for RIS OS: hapter three Reference

29 Vector Tool The Vector Tool lets you create a Vector! In abriolet, this consists of a Vector Point (see separate Vector Point Tool) joined to any other sort of Point with a Line Segment. Such a construction has the property that the Vector Point end of the Vector will always stay in exactly the same position relative to the other end (in both distance and direction). In operation, the Vector Point Tool is similar in operation to the Line Segment Tool: press the left hand mouse button, move the mouse and then release. If you click near to a Point, then this will become one end of the Vector, otherwise another Point will be created automatically. However, unlike the Line Segment Tool, if you release the mouse near a Point, the Vector will not attach itself to it. Lines,, and E are all Vectors. Line E is a Line Segment. Mathematical efinition Vector Sum Tool Mathematical efinition E If Point is moved, the whole shape is preserved and moves with. If any other Point is moved the shape is altered accordingly. Geometrically, a vector is something with both distance and direction. So, for instance, a vector is a measure of the relative position of two points. See Vector Sum Tool for some more detail. The Vector Sum Tool can be used to create the Vector that is the sum (also known as the resultant) of two other Vectors. To use this tool, first select two Vectors. This can be done either by clicking on a Vector or by clicking on a Vector Point (the endpoint of a Vector). Then click on a Point that you wish the new Vector to start at. If you do not click on a Point then one will be created automatically. vector can be thought of as an instruction to move a certain distance in a certain direction. The resultant (or sum) of two vectors is defined to be a single distance and direction instruction that will move you to the same place as if you had carried out the instructions relating to the two original vectors one after the other. i.e. the route taken does not matter; vectors are added as the crow flies. hapter three Reference 23

30 opy Vector Tool The opy Vector Tool can be used in two related ways. Firstly it can be used to simply make a copy of a Vector but with a different starting point. Secondly it can be used to make a Vector that can be defined by any two Points. To use this Tool, first click either on a Line Segment (which may be a Vector) or on any two Points (these can be the start and end Points of a Vector). Then click on a Point that you wish the new Vector to start at (if you do not click on a Point, a new one will be created automatically). is a Vector. is a Vector opy of. Hence is a parallelogram. 2 If Point is moved, Point also moves. If Point is moved, Point also moves. If Point is moved, Point also moves. is a Line Segment. is a Vector opy of. Hence is a parallelogram. If Point is moved, Point also moves. If Point is moved, Point also moves. If Point is moved, Point also moves. ddition Tool This Tool can be used to add up any two Measurements and then display the result. This can then be treated like any other Measurement; it can be moved around and it can have other calculations performed upon it. The same is true of the other Tools on this page. To use it, simply click on any two Measurements. Subtraction Tool This Tool can be used to find the difference between any two Measurements. To use it, simply click on any two Measurements. Multiplication Tool This Tool can be used to find the product of any two Measurements. To use it, simply click on any two Measurements. Square Tool This Tool can be used to find the square of any Measurement. To use it, simply click on any Measurement. Square Root Tool This Tool can be used to find the square root of any Measurement. To use it, simply click on any Measurement. 24 abriolet for RIS OS: hapter three Reference

31 X oordinate Tool This Tool can be used to display the ordinate of any Point. To use it, simply click on a Point. Y oordinate Tool This Tool can be used to display the abscissa of any Point. To use it, simply click on a Point. hapter three Reference 25